Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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20 I Fundamental Concepts<br />
We wish to prove that f is constant on every open set of U. Indeed, let U ∈ U and<br />
x ∈ U be given and suppose that f (x)=0; then [0,x] is covered by a f<strong>in</strong>ite subcover<strong>in</strong>g<br />
V ⊂ U. But for every y ∈ U, the <strong>in</strong>terval [0,y] is covered by a f<strong>in</strong>ite family<br />
of open <strong>in</strong>tervals of U (that is to say, by V ∪{U}) mean<strong>in</strong>g thereby that f (y)=0.<br />
This shows that f equals either 0 or 1 on the entire open <strong>in</strong>terval U. It follows that<br />
f is cont<strong>in</strong>uous; moreover, s<strong>in</strong>ce I =[0,1] is connected, by Theorem (I.1.12), f is<br />
constant; s<strong>in</strong>ce f (0)=0, we have f (1)=0, <strong>in</strong> other words, I is covered by a f<strong>in</strong>ite<br />
subcover<strong>in</strong>g of U. �<br />
We note that, s<strong>in</strong>ce every closed <strong>in</strong>terval [a,b] where a < b (bounded) is homeomorphic<br />
to the unit <strong>in</strong>terval [0,1], every bounded and closed <strong>in</strong>terval of R is compact.<br />
We wish to prove that a f<strong>in</strong>ite product of compact spaces is a compact space;<br />
this result is an immediate consequence of the follow<strong>in</strong>g theorem:<br />
(I.1.31) Theorem. Let B and C be compact subspaces of the topological spaces X<br />
and Y, respectively, and let U = {A j | j ∈ J} be an open cover<strong>in</strong>g of B × C. Then<br />
there are two open sets U ⊂ X and V ⊂ Y such that<br />
B ×C ⊂ U ×V ⊂ X ×Y<br />
and U ×V is covered by a f<strong>in</strong>ite number of elements of U.<br />
Proof. The method for prov<strong>in</strong>g this theorem is the one used <strong>in</strong> Theorem (I.1.29) to<br />
show the existence of two open sets U and V which conta<strong>in</strong> f −1 (y1) and f −1 (y2),<br />
respectively. We leave to the reader the task of supply<strong>in</strong>g the details needed for this<br />
proof. �<br />
(I.1.32) Corollary. Let X1,...,Xn be compact spaces; then X1 ×...×Xn is compact.<br />
Proof. If n = 2 we have the previous theorem where B = X = X1 and C = Y = X2;<br />
thereafter, we proceed by <strong>in</strong>duction. �<br />
(I.1.33) Corollary. Let B and C be compact subspaces of the topological spaces X<br />
and Y respectively. Let W be open <strong>in</strong> X ×Y such that B ×C ⊂ W . Then there are<br />
open sets U ⊂ X and V ⊂ Y such that<br />
B ×C ⊂ U ×V ⊂ W.<br />
Proof. It is enough to set U = {W} <strong>in</strong> Theorem (I.1.31). �<br />
(I.1.34) Corollary. Let X be a Hausdorff space and let B and C be disjo<strong>in</strong>t compact<br />
subspaces of X. Then there are two disjo<strong>in</strong>t open sets U and V of X with B ⊂ U and<br />
C ⊂ V.<br />
Proof. We beg<strong>in</strong> by not<strong>in</strong>g that if<br />
ΔX = {(x,x) ∈ X × X}